Bargaining foundation for ratio equilibrium in public‐good economies

• Author: Anne van den Nouweland,Agnieszka Rusinowska
• DOI: http://doi.org/10.1111/jpet.12367
• Date: 01 April 2020
• Published date: 01 April 2020

Received: 12 February 2018

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Revised: 23 December 2018

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Accepted: 12 February 2019

DOI: 10.1111/jpet.12367

ORIGINAL ARTICLE

Bargaining foundation for ratio equilibrium in

public‐good economies

Anne van den Nouweland

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Agnieszka Rusinowska

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1

Department of Economics, University of

Oregon, Eugene, Oregon

2

Centre d’Economie de la Sorbonne—

CNRS, Paris School of Economics,

Paris, France

Correspondence

Anne van den Nouweland, Department

of Economics, University of Oregon,

Eugene, OR 97403‐1285.

Email: annev@uoregon.edu

Abstract

We provide a bargaining foundation for the concept of

ratio equilibrium in public‐good economies. We define a

bargaining game of alternating offers, in which players

bargain to determine their cost shares of public‐good

production and a level of public good. We study the

stationary subgame perfect equilibrium (SSPE) without

delay of the bargaining game. We demonstrate that

when the players are perfectly patient, they are

indifferent between the equilibrium offers of all players.

We also show that every SSPE without delay in which

the ratios offered by all players are the same leads to a

ratio equilibrium. In addition, we demonstrate that all

equilibrium ratios are offered by the players at some

SSPE without delay. We use these results to discuss the

case when the assumption of perfectly patient players is

relaxed and the cost of delay vanishes.

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INTRODUCTION

Ratio equilibrium in a public‐good economy is defined by a profile of cost ratios (one for each

player) and a configuration consisting of a level of private‐good consumption for every player

and a level of public good for the economy. A player’s cost ratio specifies that player’s share of

the cost of public‐good production for any level of public good, and in a ratio equilibrium all

players agree on the level of public good when each chooses their

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utility‐maximizing

consumption of public good within the budget constraint created by their ratio.

The current paper provides a bargaining foundation for ratio equilibrium. This addresses

how the players in a public‐good economy can negotiate over cost ratios and a level of public

good to be commonly provided and paid for. We provide a simple and natural bargaining

J Public Econ Theory. 2020;22:302–319.wileyonlinelibrary.com/journal/jpet302

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© 2019 Wiley Periodicals, Inc.

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We use the gender‐neutral “they”and “their”for both singular and plural pronouns.

procedure that implements ratio equilibrium in a subgame perfect Nash equilibrium. Because

the determination of a level of public good and the sharing of the cost of its production go

hand in hand, the bargaining procedure must and does determine both. We define an n‐player

bargaining procedure in which players take turns proposing cost ratios, and if and when a

proposal is approved by all remaining players, the last player to approve chooses a level of

public good. This reflects the spirit of ratio equilibrium, in which the players agree on their

most preferred level of public good given their budget set as determined by their cost ratios.

We consider stationary subgame perfect equilibria (SSPE) without delay of the bargaining

game. We demonstrate that when the players are perfectly patient, they are indifferent between

the equilibrium offers of all players. We also show that every SSPE without delay in which the

ratios offered by all players are the same leads to a ratio equilibrium. In addition, we

demonstrate that all equilibrium ratios are offered by the players at some SSPE without delay.

We use these results to discuss the case when the assumption of perfectly patient players is

relaxed and the cost of delay vanishes.

1.1

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Related literature

Ratio equilibrium was introduced in Kaneko (1977a) as an alternative to Lindahl equilibrium.

Whereas Lindahl equilibrium is based on players paying personalized prices per unit of the

public good, ratio equilibrium is based on players paying personalized shares of the cost of

public‐good production. Lindahl equilibrium is commonly accepted to have been introduced

by Lindahl (1919) and later formalized by Foley (1970), Johansen (1963), and Samuelson

(1954). However, van den Nouweland, Tijs, and Wooders (2002) use an axiomatic approach

to demonstrate that not Lindahl equilibrium, but ratio equilibrium accurately represents

the cost‐share ideas expressed in Lindahl (1919).

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Kaneko (1977a) addresses the existence of

a ratio equilibrium and Kaneko (1977a, 1977b) address the relationship between ratio

equilibrium and the core of a voting game, in which the ratios are exogenous or endogenous,

respectively.

The bargaining procedure that we propose is in the spirit of the bargaining game with

alternating offers (Fishburn & Rubinstein, 1982; Osborne & Rubinstein, 1990; Rubinstein,

1982) and the unanimity game with nplayers (Binmore, 1985; Chatterjee & Sabourian, 2000).

One of the issues in this context is a multiplicity of subgame perfect equilibria (SPE) when the

number of bargainers is larger than 2. Several authors investigate n‐person bargaining

procedures that lead to the uniqueness of SPE, for example, exit games (Chae & Yang, 1988,

1994; Krishna & Serrano, 1996; Yang, 1992), sequential share bargaining based on the analysis

of one‐dimensional bargaining problems (Herings & Predtetchinski, 2010, 2012). Our

bargaining game differs from the existing bargaining models in that the players have to

determine cost ratios and a level of public good, rather than determining a division of

available units of a good.

Several authors propose the Nash implementation of Lindahl allocations and of the ratio

correspondence, see Tian (1989), Corchón and Wilkie (1996), and Duggan (2002). The current

paper uses a different approach because we are interested in subgame perfect implementation

of the ratio equilibrium. The current paper is more related to Dávila, Eeckhout, and Martinelli

(2009), who investigate a two‐agent bargaining procedure whose SPE converge to Lindahl

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van den Nouweland (2015) describes how the literature developed from Lindahl (1919) to Lindahl equilibrium and discusses the relation between ratio

equilibrium and the ideas in Lindahl (1919).

VAN DEN NOUWELAND AND RUSINOWSKA

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